Given a variety \(X\) , we work with the bounded constructible derived category, \(D^b(X,\mathbb {Q})\)  (see [1]} for the basics and references) and Saito's bounded derived category of algebraic mixed Hodge modules which rational structure, \(D^bMHM_{alg}(X)\)  (see [2]} for the foundations and [3]} for basics and references). There is an exact functor \(\textup {rat}\colon D^bMHM_{alg}(X) \rightarrow D^b(X, \mathbb {Q})\)  which takes the standard \(t\) -structure on \(D^bMHM_{alg}(X)\)  to the middle perversity \(t\) -structure on \(D^b(X, \mathbb {Q})\) . An important fact is that a splitting of an object \(K \in D^bMHM_{alg}(X)\)  induces a splitting of the object \(\textup {rat}(K) \in D^b(X,\mathbb {Q})\) .
